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Bernaschi M, González-Adalid Pemartín I, Martín-Mayor V, Parisi G. The quantum transition of the two-dimensional Ising spin glass. Nature 2024; 631:749-754. [PMID: 38987607 PMCID: PMC11269196 DOI: 10.1038/s41586-024-07647-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2023] [Accepted: 06/03/2024] [Indexed: 07/12/2024]
Abstract
Quantum annealers are commercial devices that aim to solve very hard computational problems1, typically those involving spin glasses2,3. Just as in metallurgic annealing, in which a ferrous metal is slowly cooled4, quantum annealers seek good solutions by slowly removing the transverse magnetic field at the lowest possible temperature. Removing the field diminishes the quantum fluctuations but forces the system to traverse the critical point that separates the disordered phase (at large fields) from the spin-glass phase (at small fields). A full understanding of this phase transition is still missing. A debated, crucial question regards the closing of the energy gap separating the ground state from the first excited state. All hopes of achieving an exponential speed-up, compared to classical computers, rest on the assumption that the gap will close algebraically with the number of spins5-9. However, renormalization group calculations predict instead that there is an infinite-randomness fixed point10. Here we solve this debate through extreme-scale numerical simulations, finding that both parties have grasped parts of the truth. Although the closing of the gap at the critical point is indeed super-algebraic, it remains algebraic if one restricts the symmetry of possible excitations. As this symmetry restriction is experimentally achievable (at least nominally), there is still hope for the quantum annealing paradigm11-13.
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Affiliation(s)
| | | | - Víctor Martín-Mayor
- Departamento de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain
| | - Giorgio Parisi
- Dipartimento di Fisica, Sapienza Università di Roma, Rome, Italy
- Nanotec-Rome unit, CNR, Rome, Italy
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2
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Anfray V, Chatelain C. Numerical evidence of a universal critical behavior of two-dimensional and three-dimensional random quantum clock and Potts models. Phys Rev E 2023; 108:014124. [PMID: 37583146 DOI: 10.1103/physreve.108.014124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2022] [Accepted: 06/26/2023] [Indexed: 08/17/2023]
Abstract
The random quantum q-state clock and Potts models are studied in two and three dimensions. The existence of Griffiths phases is tested in the two-dimensional case with q=6 by sampling the integrated probability distribution of local susceptibilities of the equivalent McCoy-Wu three-dimensional classical models with Monte Carlo simulations. For the random Potts model, numerical evidence of the existence of Griffiths phases is given and the finite-size effects are analyzed. For the clock model, the data also suggest the existence of a Griffiths phase but with much larger finite-size effects. The critical point of the random quantum clock model is then studied with the Strong-Disorder Renormalization Group. Evidence is given that, at strong enough disorder, this critical behavior is governed by the same infinite-disorder fixed point as the Potts model, for all the number of states q considered. At weak disorder, our renormalization group method becomes unstable and does not allow us to make conclusions.
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Affiliation(s)
- Valentin Anfray
- Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France
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3
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Sinha A. Development of research network on Quantum Annealing Computation and Information using Google Scholar data. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20210413. [PMID: 36463919 DOI: 10.1098/rsta.2021.0413] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2022] [Accepted: 09/08/2022] [Indexed: 06/17/2023]
Abstract
We build and analyse the network of 100 top-cited nodes (research papers and books from Google Scholar; the strength or citation of the nodes range from about 44 000 up to 100) starting in early 1980 until last year. These searched publications (papers and books) are based on Quantum Annealing Computation and Information categorized into four different sets: (A) Quantum/Transverse Field Spin Glass Model, (B) Quantum Annealing, (C) Quantum Adiabatic Computation and (D) Quantum Computation Information in the title or abstract of the searched publications. We fitted the growth in the annual number of publication ([Formula: see text]) in each of these four categories, A-D, to the form [Formula: see text] where [Formula: see text] denotes the time in years. We found the scaling time [Formula: see text] to be of the order of about 10 years for categories A and C, whereas [Formula: see text] is of the order of about 5 years for categories B and D. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Antika Sinha
- Department of Computer Science, Asutosh College, Kolkata, West Bengal 700026, India
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4
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Singh RRP, Young AP. Critical and Griffiths-McCoy singularities in quantum Ising spin glasses on d-dimensional hypercubic lattices: A series expansion study. Phys Rev E 2017; 96:022139. [PMID: 28950636 DOI: 10.1103/physreve.96.022139] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2017] [Indexed: 06/07/2023]
Abstract
We study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
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Affiliation(s)
- R R P Singh
- Department of Physics, University of California, Davis, Davis, California 95616, USA
| | - A P Young
- Department of Physics, University of California, Santa Cruz, Santa Cruz, California 95064, USA
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5
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Khandelwal A, Chattopadhyay MK, Roy SB. Magnetotransport and magnetothermal properties of the ternary intermetallic compound TbFe2Al10. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:356001. [PMID: 27385638 DOI: 10.1088/0953-8984/28/35/356001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We have studied the temperature and field dependences of electrical resistivity and heat capacity of TbFe2Al10, and have also complimented the above studies with low field magnetization measurements. In zero magnetic field, TbFe2Al10 exhibits paramagnetic (PM) to ferrimagnetic (Ferri-I) and Ferri-I to antiferromagnetic (AFM) phase transitions below 17.6 and 10 K respectively. We have found that the electrical resistivity of TbFe2Al10 exhibits a sharp rise across the PM to Ferri-I phase transition in this compound. Our analysis indicates that this sharp rise of electrical resistivity is related to the formation of new zone boundaries (across the PM to Ferri-I phase transition) that reduce the area of the Fermi surface. We have found that TbFe2Al10 exhibits large magnetoresistance (MR) below 100 K. Overall, the MR behaviour of TbFe2Al10 below 17.6 K in different magnetic fields reveals strong competition between AFM and ferromagnetic (FM) correlations, which seems to be quite intrinsic to the magnetic structure of the compound. Our analysis indicates that the large MR and magnetocaloric effect persisting deep inside the PM regime of TbFe2Al10 is mainly related to the presence of FM spin fluctuations and the formation of a Griffiths like (GL) phase consisting of FM clusters within the PM regime. The formation of the GL phase may be mediated by the static crystal defects in the midst of the competing inter and intra layer magnetic interactions.
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Affiliation(s)
- Ashish Khandelwal
- Magnetic & Superconducting Materials Section, Raja Ramanna Centre for Advanced Technology, Indore 452013, India
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6
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Gingras MJP, Henelius P. Collective Phenomena in the LiHoxY1−xF4Quantum Ising Magnet: Recent Progress and Open Questions. ACTA ACUST UNITED AC 2011. [DOI: 10.1088/1742-6596/320/1/012001] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Mohan P, Goldbart PM, Narayanan R, Toner J, Vojta T. Anomalously elastic intermediate phase in randomly layered superfluids, superconductors, and planar magnets. PHYSICAL REVIEW LETTERS 2010; 105:085301. [PMID: 20868107 DOI: 10.1103/physrevlett.105.085301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2010] [Revised: 07/13/2010] [Indexed: 05/29/2023]
Abstract
We show that layered quenched randomness in planar magnets leads to an unusual intermediate phase between the conventional ferromagnetic low-temperature and paramagnetic high-temperature phases. In this intermediate phase, which is part of the Griffiths region, the spin-wave stiffness perpendicular to the random layers displays anomalous scaling behavior, with a continuously variable anomalous exponent, while the magnetization and the stiffness parallel to the layers both remain finite. Analogous results hold for superfluids and superconductors. We study the two phase transitions into the anomalous elastic phase, and we discuss the universality of these results, and implications of finite sample size as well as possible experiments.
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Affiliation(s)
- Priyanka Mohan
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
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8
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Ancona-Torres C, Silevitch DM, Aeppli G, Rosenbaum TF. Quantum and classical glass transitions in LiHoxY1-xF4. PHYSICAL REVIEW LETTERS 2008; 101:057201. [PMID: 18764428 DOI: 10.1103/physrevlett.101.057201] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2008] [Revised: 04/02/2008] [Indexed: 05/26/2023]
Abstract
When performed in the proper low-field, low-frequency limits, measurements of the dynamics and the nonlinear susceptibility in the model Ising magnet in a transverse field LiHo(x)Y(1-x)F(4) prove the existence of a spin-glass transition for x=0.167 and 0.198. The classical behavior tracks for the two concentrations, but the behavior in the quantum regime at large transverse fields differs because of the competing effects of quantum entanglement and random fields.
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Affiliation(s)
- C Ancona-Torres
- The James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL 60637, USA
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9
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Hoyos JA, Vojta T. Theory of smeared quantum phase transitions. PHYSICAL REVIEW LETTERS 2008; 100:240601. [PMID: 18643562 DOI: 10.1103/physrevlett.100.240601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2008] [Indexed: 05/26/2023]
Abstract
We present an analytical strong-disorder renormalization group theory of the quantum phase transition in the dissipative random transverse-field Ising chain. For Ohmic dissipation, we solve the renormalization flow equations analytically, yielding asymptotically exact results for the low-temperature properties of the system. We find that the interplay between quantum fluctuations and Ohmic dissipation destroys the quantum critical point by smearing. We also determine the phase diagram and the behavior of observables in the vicinity of the smeared quantum phase transition.
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Affiliation(s)
- José A Hoyos
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
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10
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Tabei SMA, Gingras MJP, Kao YJ, Stasiak P, Fortin JY. Induced random fields in the LiHoxY1-xF4 quantum Ising magnet in a transverse magnetic field. PHYSICAL REVIEW LETTERS 2006; 97:237203. [PMID: 17280239 DOI: 10.1103/physrevlett.97.237203] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2006] [Indexed: 05/13/2023]
Abstract
The LiHoxY1-xF4 magnetic material in a transverse magnetic field Bx x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in a random disordered system. We show that the Bx-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline symmetries, generates, via the predominant dipolar interactions between Ho3+ ions, random fields along the Ising z direction. This identifies LiHoxY1-xF4 in Bx as a new random field Ising system. The random fields explain the rapid decrease of the critical temperature in the diluted ferromagnetic regime and the smearing of the nonlinear susceptibility at the spin-glass transition with increasing Bx and render the Bx-induced quantum criticality in LiHoxY1-xF4 likely inaccessible.
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Affiliation(s)
- S M A Tabei
- Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada
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11
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Schehr G, Rieger H. Strong-disorder fixed point in the dissipative random transverse-field Ising model. PHYSICAL REVIEW LETTERS 2006; 96:227201. [PMID: 16803340 DOI: 10.1103/physrevlett.96.227201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2005] [Indexed: 05/10/2023]
Abstract
The interplay between disorder, quantum fluctuations, and dissipation is studied in the random transverse Ising chain coupled to a dissipative Ohmic bath with a real space renormalization group. A typically very large length scale L* is identified above which the physics of frozen clusters dominates. Below L* a strong-disorder fixed point determines scaling at a pseudocritical point. In a Griffiths-McCoy region frozen clusters produce already a finite magnetization resulting in a classical low temperature behavior of the susceptibility and specific heat. These override the confluent singularities that are characterized by a continuously varying exponent z and are visible above a temperature T*approximately L*-z.
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Affiliation(s)
- Grégory Schehr
- Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany
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12
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Vojta T. Rare region effects at classical, quantum and nonequilibrium phase transitions. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/0305-4470/39/22/r01] [Citation(s) in RCA: 316] [Impact Index Per Article: 17.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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13
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Magen C, Algarabel PA, Morellon L, Araújo JP, Ritter C, Ibarra MR, Pereira AM, Sousa JB. Observation of a Griffiths-like phase in the magnetocaloric compound Tb5Si2Ge2. PHYSICAL REVIEW LETTERS 2006; 96:167201. [PMID: 16712265 DOI: 10.1103/physrevlett.96.167201] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2005] [Indexed: 05/09/2023]
Abstract
The onset of a Griffiths-like phase has been observed in Tb5Si2Ge2 (TC=110 K) by means of magnetic susceptibility and small-angle neutron scattering experiments. We show the growth of a ferromagnetic cluster system characterized by an inverse susceptibility exponent lower than unity at TC<T<TG approximately 200 K. We suggest that the Griffiths-like state is originated by local disorder within the crystallographic structure, stabilized and enhanced by competing intralayer and interlayer magnetic interactions. Both factors thus promote segregation of nanometric regions with ferromagnetic interactions.
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Affiliation(s)
- C Magen
- Instituto de Ciencia de Materiales de Aragón, Universidad de Zaragoza and Consejo Superior de Investigaciones Científicas, 50009 Zaragoza, Spain
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14
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Carpentier D, Pujol P, Giering KU. Random quantum Ising chains with competing interactions. Phys Rev E 2006; 72:066101. [PMID: 16486004 DOI: 10.1103/physreve.72.066101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2005] [Indexed: 11/07/2022]
Abstract
In this paper we discuss the criticality of a quantum Ising spin chain with competing random ferromagnetic and antiferromagnetic couplings. Quantum fluctuations are introduced via random local transverse fields. First we consider the chain with couplings between first and second neighbors only and then generalize the study to a quantum analog of the Viana-Bray model, defined on a small world random lattice. We use the Dasgupta-Ma decimation technique, both analytically and numerically, and focus on the scaling of the lattice topology, whose determination is necessary to define any infinite disorder transition beyond the chain. In the first case, at the transition the model renormalizes towards the chain, with the infinite disorder fixed point described by Fisher. This corresponds to the irrelevance of the competition induced by the second neighbors couplings. As opposed to this case, this infinite disorder transition is found to be unstable towards the introduction of an arbitrary small density of long range couplings in the small world models.
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Affiliation(s)
- David Carpentier
- Laboratoire de Physique de l'Ecole Normale Supérieure de Lyon, 46, Allée d'Italie, 69007 Lyon, France
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15
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Schechter M, Stamp PCE. Significance of the hyperfine interactions in the phase diagram of LiHoxY1-xF4. PHYSICAL REVIEW LETTERS 2005; 95:267208. [PMID: 16486398 DOI: 10.1103/physrevlett.95.267208] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2005] [Indexed: 05/06/2023]
Abstract
We consider the quantum magnet at LiHo(x)Y(1-x)F(4) at x = 0.167. Experimentally the spin glass to paramagnet transition in this system was studied as a function of the transverse magnetic field and temperature, showing peculiar features: for example, (i) the spin glass order is destroyed much faster by thermal fluctuations than by the transverse field; and (ii) the cusp in the nonlinear susceptibility signaling the glass state decreases in size at lower temperature. Here we show that the hyperfine interactions of the Ho atom must dominate in this system, and that along with the transverse inter-Ho dipolar interactions they dictate the structure of the phase diagram. The experimental observations are shown to be natural consequences of this.
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Affiliation(s)
- M Schechter
- Department of Physics & Astronomy and Pacific Institute for Theoretical Physics, University of British Columbia, Vancouver, Canada
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16
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Dobrosavljević V, Miranda E. Absence of conventional quantum phase transitions in itinerant systems with disorder. PHYSICAL REVIEW LETTERS 2005; 94:187203. [PMID: 15904407 DOI: 10.1103/physrevlett.94.187203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2004] [Indexed: 05/02/2023]
Abstract
Effects of disorder are examined in itinerant systems close to quantum critical points. We argue that spin fluctuations due to the long-range part of the RKKY interactions generically induce non-Ohmic dissipation due to rare disorder configurations. This dissipative mechanism is found to destabilize quantum Griffiths phase behavior in itinerant systems with arbitrary symmetry of the order parameter, leading to the formation of a "cluster glass" phase preceding uniform ordering.
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Affiliation(s)
- V Dobrosavljević
- Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA
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17
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Vojta T. Disorder-induced rounding of certain quantum phase transitions. PHYSICAL REVIEW LETTERS 2003; 90:107202. [PMID: 12689028 DOI: 10.1103/physrevlett.90.107202] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2002] [Indexed: 05/24/2023]
Abstract
We study the influence of quenched disorder on quantum phase transitions in systems with overdamped dynamics. For Ising order-parameter symmetry disorder destroys the sharp phase transition by rounding because a static order parameter can develop on rare spatial regions. This leads to an exponential dependence of the order parameter on the coupling constant. At finite temperatures the static order on the rare regions is destroyed. This restores the phase transition and leads to a double-exponential relation between critical temperature and coupling strength. We discuss the behavior based on Lifshitz-tail arguments and illustrate the results by simulations of a model system.
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Affiliation(s)
- Thomas Vojta
- Department of Physics, University of Missouri-Rolla, Rolla, Missouri 65409, USA
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18
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Vajk OP, Greven M. Quantum versus geometric disorder in a two-dimensional Heisenberg antiferromagnet. PHYSICAL REVIEW LETTERS 2002; 89:177202. [PMID: 12398702 DOI: 10.1103/physrevlett.89.177202] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2002] [Indexed: 05/24/2023]
Abstract
We present a numerical study of the spin-1/2 bilayer Heisenberg antiferromagnet with random interlayer dimer dilution. From the temperature dependence of the uniform susceptibility and a scaling analysis of the spin correlation length we deduce the ground state phase diagram as a function of nonmagnetic impurity concentration p and bilayer coupling g. At the site percolation threshold, there exists a multicritical point at small but nonzero bilayer coupling g(m)=0.15(3). The magnetic properties of the single-layer material La(2)Cu(1-p)(Zn,Mg)(p)O4 near the percolation threshold appear to be controlled by the proximity to this new quantum critical point.
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Affiliation(s)
- O P Vajk
- Department of Physics, Stanford University, Stanford, California 94305, USA
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19
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Marqués MI, Gonzalo JA. Dynamic scaling in diluted systems: deactivation through thermal dilution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056114. [PMID: 11414968 DOI: 10.1103/physreve.63.056114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2000] [Indexed: 05/23/2023]
Abstract
Activated scaling is confirmed to hold in transverse field-induced phase transitions of randomly diluted Ising systems. Quantum Monte Carlo calculations have been made not just at the percolation threshold (pc) but also well below and above it. We follow the evolution of the activated scaling at zero temperature in the phase transition from ferromagnetic to quantum Griffiths phase (p>pc) at the phase boundary (p=pc) and for transitions inside the nonferromagnetic quantum Griffiths phase (p<pc). A novel deactivation phenomenon inside the nonferromagnetic Griffiths-McCoy phase (p<pc) is observed using a thermal (in contrast to random) dilution of the system.
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Affiliation(s)
- M I Marqués
- Departamento de Física de Materiales C-IV, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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20
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Rieger H, Young AP. Griffiths singularities in the disordered phase of a quantum Ising spin glass. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:3328-3335. [PMID: 9986232 DOI: 10.1103/physrevb.54.3328] [Citation(s) in RCA: 64] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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